|
%I #47 Mar 23 2021 10:41:45
%S 1,4,4,12,4,12,12,36,4,12,12,36,12,36,36,108,4,12,12,36,12,36,36,108,
%T 12,36,36,108,36,108,108,324,4,12,12,36,12,36,36,108,12,36,36,108,36,
%U 108,108,324,12,36,36,108,36,108,108,324,36,108,108,324,108,324,324,972,4
%N First differences of A147562.
%C Bisection of A323651. - _Omar E. Pol_, Mar 04 2019
%D D. Singmaster, On the cellular automaton of Ulam and Warburton, M500 Magazine of the Open University, #195 (December 2003), pp. 2-7.
%D S. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962.
%H N. J. A. Sloane, <a href="/A147582/b147582.txt">Table of n, a(n) for n = 1..10000</a>
%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4*3^(wt(n-1)-1) for n >= 2.]
%H David Applegate, <a href="/A139250/a139250.anim.html">The movie version</a>
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polca004.jpg">Illustration of initial terms (Fig. 1: one-step rook)</a>, <a href="http://www.polprimos.com/imagenespub/polca006.jpg">(Fig. 2: one-step bishop)</a>, <a href="http://www.polprimos.com/imagenespub/polca008.jpg">(Fig. 3: overlapping squares)</a>, <a href="http://www.polprimos.com/imagenespub/polca010.jpg">(Fig. 4: overlapping X-toothpicks)</a>, 2009
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polca002.jpg">Illustration of initial terms of A139251, A160121, A147582 (Overlapping figures)</a>, 2009
%H D. Singmaster, <a href="/A079314/a079314.pdf">On the cellular automaton of Ulam and Warburton</a>, 2003 [Cached copy, included with permission]
%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H N. J. A. Sloane, <a href="https://www.youtube.com/watch?v=9ogbsh8KuEM">Exciting Number Sequences</a> (video of talk), Mar 05 2021
%F a(1) = 1; for n > 1, a(n) = 4*3^(wt(n-1)-1) where wt() = A000120(). - _R. J. Mathar_, Apr 30 2009
%F This formula is (essentially) given by Singmaster. - _N. J. A. Sloane_, Aug 06 2009
%F G.f.: x + 4*x*(Product_{k >= 0} (1 + 3*x^(2^k)) - 1)/3. - _N. J. A. Sloane_, Jun 10 2009
%e From _Omar E. Pol_, Jun 14 2009: (Start)
%e When written as a triangle:
%e .1;
%e .4;
%e .4,12;
%e .4,12,12,36;
%e .4,12,12,36,12,36,36,108;
%e .4,12,12,36,12,36,36,108,12,36,36,108,36,108,108,324;
%e .4,12,12,36,12,36,36,108,12,36,36,108,36,108,108,324,12,36,36,108,36,108,...
%e The rows converge to A161411. (End)
%p A000120 := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end: wt := A000120; A147582 := n-> if n <= 1 then n else 4*3^(wt(n-1)-1); fi; [seq(A147582(n),n=0..1000)]; # _N. J. A. Sloane_, Apr 07 2010
%t s = Plus @@ Flatten@ # & /@ CellularAutomaton[{686, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, 200]; f[n_] = If[n == 0, 1, s[[n + 1]] - s[[n]]]; Array[f, 120, 0] (* _Michael De Vlieger_, Apr 09 2015, after _Nadia Heninger_ and _N. J. A. Sloane_ at A147562 *)
%Y Cf. A147562, A147610 (the sequence divided by 4), A048881, A000120.
%Y Cf. A000079, A161411, A151779, A139250.
%Y Cf. A048883, A139251, A160121, A162349. [_Omar E. Pol_, Nov 02 2009]
%Y Cf. A323651.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Apr 29 2009
%E Extended by _R. J. Mathar_, Apr 30 2009
|
